1

Intractable Problems

Time-Bounded Turing Machines

Classes P and NPPolynomial-Time Reductions

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Time-Bounded TM’s

A Turing machine that, given an input of length n, always halts within T(n) moves is said to be T(n)-time bounded. The TM can be multitape. Sometimes, it can be nondeterministic.

The deterministic, multitape case corresponds roughly to “an O(T(n)) running-time algorithm.”

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The class P

If a DTM M is T(n)-time bounded for some polynomial T(n), then we say M is polynomial-time (“polytime ”) bounded.

And L(M) is said to be in the class P. Important point: when we talk of P, it

doesn’t matter whether we mean “by a computer” or “by a TM” (next slide).

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Polynomial Equivalence of Computers and TM’s

A multitape TM can simulate a computer that runs for time O(T(n)) in at most O(T2(n)) of its own steps.

If T(n) is a polynomial, so is T2(n).

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Examples of Problems in P

Is w in L(G), for a given CFG G? Input = w. Use CYK algorithm, which is O(n3).

Is there a path from node x to node y in graph G? Input = x, y, and G. Use Dijkstra’s algorithm, which is O(n

log n) on a graph of n nodes and arcs.

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Running Times Between Polynomials

You might worry that something like O(n log n) is not a polynomial.

However, to be in P, a problem only needs an algorithm that runs in time less than some polynomial.

Surely O(n log n) is less than the polynomial O(n2).

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A Tricky Case: Knapsack

The Knapsack Problem is: given positive integers i1, i2 ,…, in, can we divide them into two sets with equal sums?

Perhaps we can solve this problem in polytime by a dynamic-programming algorithm: Maintain a table of all the differences we

can achieve by partitioning the first j integers.

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Knapsack – (2)

Basis: j = 0. Initially, the table has “true” for 0 and “false” for all other differences.

Induction: To consider ij, start with a new table, initially all false.

Then, set k to true if, in the old table, there is a value m that was true, and k is either m+ij or m-ij.

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Knapsack – (3)

Suppose we measure running time in terms of the sum of the integers, say m.

Each table needs only space O(m) to represent all the positive and negative differences we could achieve.

Each table can be constructed in time O(n).

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Knapsack – (4)

Since n < m, we can build the final table in O(m2) time.

From that table, we can see if 0 is achievable and solve the problem.

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Subtlety: Measuring Input Size

“Input size” has a specific meaning: the length of the representation of the problem instance as it is input to a TM.

For the Knapsack Problem, you cannot always write the input in a number of characters that is polynomial in either the number-of or sum-of the integers.

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Knapsack – Bad Case

Suppose we have n integers, each of which is around 2n.

We can write integers in binary, so the input takes O(n2) space to write down.

But the tables require space O(n2n). They therefore require at least that

order of time to construct.

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Bad Case – (2)

Thus, the proposed “polynomial” algorithm actually takes time O(n22n) on an input of length O(n2).

Or, since we like to use n as the input size, it takes time O(n2sqrt(n)) on an input of length n.

In fact, it appears no algorithm solves Knapsack in polynomial time.

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Redefining Knapsack

We are free to describe another problem, call it Pseudo-Knapsack, where integers are represented in unary.

Pseudo-Knapsack is in P.

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The Class NP

The running time of a nondeterministic TM is the maximum number of steps taken along any branch.

If that time bound is polynomial, the NTM is said to be polynomial-time bounded.

And its language/problem is said to be in the class NP.

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Example: NP

The Knapsack Problem is definitely in NP, even using the conventional binary representation of integers.

Use nondeterminism to guess one of the subsets.

Sum the two subsets and compare.

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P Versus NP

Originally a curiosity of Computer Science, mathematicians now recognize as one of the most important open problems the question P = NP?

There are thousands of problems that are in NP but appear not to be in P.

But no proof that they aren’t really in P.

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Complete Problems

One way to address the P = NP question is to identify complete problems for NP.

An NP-complete problem has the property that if it is in P, then every problem in NP is also in P.

Defined formally via “polytime reductions.”

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Complete Problems – Intuition

A complete problem for a class embodies every problem in the class, even if it does not appear so.

Compare: PCP embodies every TM computation, even though it does not appear to do so.

Strange but true: Knapsack embodies every polytime NTM computation.

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Polytime Reductions

Goal: find a way to show problem L to be NP-complete by reducing every language/problem in NP to L in such a way that if we had a deterministic polytime algorithm for L, then we could construct a deterministic polytime algorithm for any problem in NP.

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Polytime Reductions – (2)

We need the notion of a polytime transducer – a TM that:

1. Takes an input of length n.2. Operates deterministically for some

polynomial time p(n).3. Produces an output on a separate

output tape. Note: output length is at most p(n).

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Polytime Transducer

state

ninput

scratchtapes

output < p(n)

Remember: important requirementis that time < p(n).

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Polytime Reductions – (3)

Let L and M be langauges. Say L is polytime reducible to M if

there is a polytime transducer T such that for every input w to T, the output x = T(w) is in M if and only if w is in L.

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Picture of Polytime Reduction

T

in L

notin L

in M

not in M

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NP-Complete Problems

A problem/language M is said to be NP-complete if for every language L in NP, there is a polytime reduction from L to M.

Fundamental property: if M has a polytime algorithm, then L also has a polytime algorithm. I.e., if M is in P, then every L in NP is also

in P, or “P = NP.”

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The Plan

NP

SAT

All of NP polytimereduces to SAT, whichis therefore NP-complete

3-SAT

SAT polytimereduces to3-SAT

3-SAT polytime reducesto many other problems;they’re all NP-complete

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Proof That Polytime Reductions “Work”

Suppose M has an algorithm of polynomial time q(n).

Let L have a polytime transducer T to M, taking polynomial time p(n).

The output of T, given an input of length n, is at most of length p(n).

The algorithm for M on the output of T takes time at most q(p(n)).

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Proof – (2)

We now have a polytime algorithm for L:

1. Given w of length n, use T to produce x of length < p(n), taking time < p(n).

2. Use the algorithm for M to tell if x is in M in time < q(p(n)).

3. Answer for w is whatever the answer for x is.

Total time < p(n) + q(p(n)) = a polynomial.